# REPRESENTATION THEORY - 2023/4

Module code: MATM035

## Module Overview

Symmetries are a powerful method for easily understanding properties of otherwise complicated mathematical and physical objects. Group theory is a branch of mathematics developed to understand symmetries, however it often leads to complicated abstract quantities. Group representation theory turns such abstract algebraic concepts into linear transformations of vector spaces, a much easier system to solve. In doing, representation theory can unveil deep symmetry properties of physical systems as well as leading to powerful and compact solutions to otherwise difficult and intractable problems.

### Module provider

Mathematics

### Module Leader

PRINSLOO Andrea (Maths & Phys)

### Number of Credits: 15

### ECTS Credits: 7.5

### Framework: FHEQ Level 7

### JACs code: G100

### Module cap (Maximum number of students): N/A

## Overall student workload

Independent Learning Hours: 73

Lecture Hours: 33

Tutorial Hours: 5

Guided Learning: 6

Captured Content: 33

## Module Availability

Semester 2

## Prerequisites / Co-requisites

None

## Module content

1. Symmetry and finite groups: review of group theory definitions, subgroups, cyclic groups, cosets, conjugacy elements and classes.

2. Representation theory basics: definitions, Schur's lemma, reducible and irreducible representations, orthogonality theorems, decomposition of reducible representations. Example representations of finite groups.

3. Lie groups and algebras: generators, Jacobi identity, Lie algebra, adjoint representation, exponentiation

4. SU(2): definition, representations, raising and lowering operators

5. Roots and weights: weights, roots, raising and lowering, su(3)

6. Dynkin diagrams and simple roots: positive weights, simple roots, Dynkin diagrams

7. Physical applications: quantum mechanics.

## Assessment pattern

Assessment type | Unit of assessment | Weighting |
---|---|---|

School-timetabled exam/test | In-semester test (50 mins) | 20 |

Examination | Exam (2 hrs) | 80 |

## Alternative Assessment

N/A

## Assessment Strategy

The __assessment strategy__ is designed to provide students with the opportunity to demonstrate:

· Understanding of and ability to interpret and manipulate mathematical statements.

· Subject knowledge through the recall of key definitions, theorems and their proofs.

· Analytical ability through the solution of unseen problems in the test and exam.

Thus, the __summative assessment__ for this module consists of:

· One examination at the end of the Semester; worth 80% of the module mark.

· One in-semester test; worth 20% of the module mark.

__Formative assessment and feedback__

Students receive written feedback via a number of un-assessed coursework assignments over the 11-week period. Students are then encouraged to arrange meetings with the module convener for verbal feedback.

## Module aims

- Introduce the concept of a group representation.
- Construct representations of finite groups and some simple examples of Lie groups and algebras.
- Develop the concept of Dynkin diagrams and it to construct representations of Lie algebras.
- Illustrate these concepts in physical systems.

## Learning outcomes

Attributes Developed | ||

001 | Understand the concepts, theorems and techniques of group representation theory. | K |

002 | Have a clear understanding of how to construct irreducible representations of groups. | CT |

003 | Be able to explicitly apply the theory to small order groups. | CT |

Attributes Developed

**C** - Cognitive/analytical

**K** - Subject knowledge

**T** - Transferable skills

**P** - Professional/Practical skills

## Methods of Teaching / Learning

The __learning and teaching__ strategy is designed to provide:

- A detailed introduction to the relevant group theory and representation theory
- Experience the methods used to interpret, understand and solve concrete problems, especially for small order groups

The

__learning and teaching__methods include:

- 3 x 1 hour lectures per week x 11 weeks, with black/whiteboard written notes to supplement the module notes.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

## Reading list

https://readinglists.surrey.ac.uk

Upon accessing the reading list, please search for the module using the module code: **MATM035**

## Programmes this module appears in

Programme | Semester | Classification | Qualifying conditions |
---|---|---|---|

Mathematics and Physics MPhys | 2 | Optional | A weighted aggregate mark of 50% is required to pass the module |

Mathematics and Physics MMath | 2 | Optional | A weighted aggregate mark of 50% is required to pass the module |

Mathematics with Statistics MMath | 2 | Optional | A weighted aggregate mark of 50% is required to pass the module |

Mathematics MSc | 2 | Optional | A weighted aggregate mark of 50% is required to pass the module |

Mathematics MMath | 2 | Optional | A weighted aggregate mark of 50% is required to pass the module |

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2023/4 academic year.